2WW Question \( 9,2.2 .57 \) Letitia, Brianne, and Jake met at the mall on December 31 . Letitia said that she intends to come to the mall every seventh day throughout the next year. Brianne said that she intends to be there every third day, and Jake said that he would be there every fifth day. Letitia said that she knew she would be at the mall a total of 52 days, since \( 365 \div 7 \) is 52 with a remainder of 1 . Brianne said that she'd be at the mall 121 days, since \( 365+3 \) is 121 with a remainder of 2 . Moreover, Brianne said that of those 121 days, she would expect to see Letitia 17 times, since they will both be coming every 21 days and \( 365+21 \) is 17 with a remainder of 8 . Complete parts (a) through (d) below. (a) How many days will all three friends meet at the mall in the next year? There will be \( \square \) day(s) when all three meet.

To determine how many days Letitia, Brianne, and Jake will all meet at the mall in the next year, we need to find the least common multiple (LCM) of the days each person visits the mall. 1. **Letitia visits every 7 days.** 2. **Brianne visits every 3 days.** 3. **Jake visits every 5 days.** We need to find the LCM of 7, 3, and 5. ### Step 1: Find the LCM The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers. - The prime factorization of each number is: - \( 7 = 7^1 \) - \( 3 = 3^1 \) - \( 5 = 5^1 \) ### Step 2: Calculate the LCM The LCM is found by taking the highest power of each prime that appears in the factorizations: \[ \text{LCM}(7, 3, 5) = 7^1 \times 3^1 \times 5^1 = 7 \times 3 \times 5 \] Now, let's calculate this: \[ 7 \times 3 = 21 \] \[ 21 \times 5 = 105 \] So, the LCM of 7, 3, and 5 is 105. ### Step 3: Determine how many times they meet in a year Now, we need to find out how many times 105 days fits into a year (365 days): \[ \text{Number of meetings} = \frac{365}{105} \] Calculating this gives: \[ \frac{365}{105} \approx 3.476 \] Since they can only meet a whole number of times, we take the integer part: \[ \text{Number of meetings} = 3 \] ### Conclusion Thus, there will be \( \boxed{3} \) days when all three friends meet at the mall in the next year.